L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s − 4·19-s + 20-s + 4·22-s + 4·23-s + 25-s + 6·26-s − 2·29-s − 31-s + 32-s − 2·34-s − 2·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + 4·44-s + 4·46-s − 7·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.371·29-s − 0.179·31-s + 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.603·44-s + 0.589·46-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.665781288\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.665781288\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.842952695778314240320502191189, −8.123847345984547628522517920045, −6.91870640349350855438029646474, −6.43519172564930714969372916538, −5.85074877093119953971883550530, −4.85068510919595048196425622933, −3.98223593528833352218557256642, −3.36426698589872969422763592624, −2.10692679331839958833516203378, −1.18814113043022149696660013228,
1.18814113043022149696660013228, 2.10692679331839958833516203378, 3.36426698589872969422763592624, 3.98223593528833352218557256642, 4.85068510919595048196425622933, 5.85074877093119953971883550530, 6.43519172564930714969372916538, 6.91870640349350855438029646474, 8.123847345984547628522517920045, 8.842952695778314240320502191189